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The idea that reasoning and problem solving (by human beings) are guided by experiences from situations which are similar to the current one has a long tradition in philosophy. It dates back at least to D.

This chapter serves two purposes. Firstly, we provide some background information on similarity-based reasoning and related topics. Secondly, we introduce a formal framework of (CBI) that provides the basis for the methods which are developed in subsequent chapters.

In this chapter, we adopt a constraint-based view of the CBI hypothesis, according to which the similarity of inputs imposes a constraint on the similarity of associated outcomes in the form of a lower bound. A related inference mechanism then allows for realizing CBI as a kind of constraint propagation. We also discuss representational issues and algorithms for putting the idea of within this framework into action. The chapter is organized as follows: Section 3.1 introduces the aforementioned formalization of the CBI hypothesis. A case-based inference scheme which emerges quite naturally from this formalization is proposed in Section 3.2 and further developed in Section 3.3. Case-based learning is discussed in Section 3.4. In Section 3.5, some applications of case-based inference in the context of statistics are outlined. The chapter concludes with a brief summary and some complementary remarks in Section 3.6.

The main idea of case-based inference is to exploit the information provided by the of a problem , 〉 in order to improve the prediction of an unknown outcome = ().

A close connection between fuzzy set-based (approximate reasoning) methods and the inference principle underlying similarity-based (case-based) reasoning has been pointed out recently [99, 407]. Besides, some attempts at combining case-based reasoning (or, more generally, analogical reasoning) and methods from fuzzy set theory have already been made [408], including the use of fuzzy sets for supporting the computation of similarities of situations in analogical reasoning [144], the formalization of aspects of analogical reasoning by means of similarity relations between fuzzy sets [48], the use of fuzzy set theory in case indexing and retrieval [209, 214], the case-based learning of fuzzy concepts from fuzzy examples [295], the use of fuzzy predicates in the derivation of similarities [40], and the integration of case-based and rule-based reasoning [138]. See [45, 49] for a more general framework of analogical reasoning.

In Chapter 5, it has already been shown that fuzzy rules can be modeled formally as possibility distributions constrained in terms of a combination of the membership functions which define, respectively, their antecedent and consequent part.

Early work in AI has mainly focused on formal logic as a basis of knowledge representation and has largely rejected approaches from (statistical) decision theory as being intractable and inadequate for expressing the rich structure of (human) knowledge [193].

In this book, we have developed various approaches to what we have called . The idea of CBI is to exploit experience in the form of a memory of observed cases (a case base consisting of input-output tuples) in order to predict a set of promising candidate outputs given a new query input. The corresponding inference schemes are based on suitable formalizations of the heuristic assumption that similar inputs yield similar outputs. Proceeding from a very simple, constraint-based model of this hypothesis, more sophisticated versions have been developed within different formal frameworks of approximate reasoning and reasoning under uncertainty. Let us again highlight the following properties of our approaches: